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From Metropolis to diffusions: Gibbs states and optimal scaling. (English) Zbl 1047.60065

Let ${\Pi }=\left({\pi }_{W}\left(a,\phantom{\rule{0.166667em}{0ex}}dx\right),\phantom{\rule{0.277778em}{0ex}}W\subseteq {ℤ}^{d}\phantom{\rule{4pt}{0ex}}\text{finite},a\in {ℝ}^{{ℤ}^{d}}\right)$ be a specification, that is, a consistent family of finite-volume conditional Gibbs measures for a finite-range Hamiltonian $H$. Suppose that $\xi$ is the corresponding infinite-volume Gibbs measure, let $\xi$ be translation invariant. Given a subset ${V}_{n}\subseteq {ℤ}^{d}$ of cardinality $n$ and a boundary condition $z$, let $\left({X}_{t}\left({V}_{n},z\right),\phantom{\rule{0.277778em}{0ex}}t\ge 0\right)$ be the random walk Metropolis chain for ${\pi }_{{V}_{n}}\left(z,·\right)$. It was shown by the second author and A. F. M. Smith [Stochastic Processes Appl. 49, 207-216 (1994; Zbl 0803.60067)] that ${X}_{t}\left({V}_{n},z\right)$ converges weakly to ${\pi }_{{V}_{n}}\left(z,·\right)$ as $t\to \infty$. The behaviour of the algorithm as ${V}_{n}↑{ℤ}^{d}$ is studied. In particular, choose the proposal variance ${\sigma }_{n}^{2}=l{n}^{-1}$. Under suitable assumptions on $H$ and $\xi$ it is proven that ${X}_{\left[nt\right]}\left({V}_{n},z\right)$ converges weakly as $n\to \infty$ to an infinite-dimensional diffusion ${Z}_{t}$ on a Hilbert space $E={L}^{2}\left({ℤ}^{d},\rho \right)$. The measure $\rho$ is given by $\rho \left(\left\{k\right\}\right)=\left({\sum }_{j\in {ℤ}^{d}}{exp\left(-|j|\right)\right)}^{-1}exp\left(-|k|\right)$, $Z$ solves the equation

$d{Z}_{t}=-\frac{l}{2}v\left({Z}_{t}\right)\nabla H\left({Z}_{t}\right)\phantom{\rule{0.166667em}{0ex}}dt+\sqrt{lv\left({Z}_{t}\right)}\phantom{\rule{0.166667em}{0ex}}d{B}_{t},$

driven by a Brownian motion $B$ in $E$ and with the initial condition ${Z}_{0}=\xi$ in law. The coefficient $v$ is defined in terms of the second derivative of the Hamiltonian $H$.

##### MSC:
 60J05 Discrete-time Markov processes on general state spaces 65C05 Monte Carlo methods 60J22 Computational methods in Markov chains
##### Keywords:
Markov chain Monte Carlo; Gibbs measures